The connected Isentropes conjecture in a space of quartic polynomials

Pages: 139 - 175, Volume 19, Issue 1, September 2007

doi:10.3934/dcds.2007.19.139       Abstract        Full Text (562.0K)       Related Articles

Anca Radulescu - Department of Applied Mathematics, 526 UCB, University of Colorado, Boulder, CO 80309-0526, United States (email)

Abstract: This note is a shortened version of my dissertation paper, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the parameters for which critical orbits are periodic, and I called such curves left and right bones. Using quasiconformal surgery methods and rigidity, I showed that the bones are simple smooth arcs that join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of the maps evolve along the bones. The behavior of the topological entropy function of the polynomials in my family is closely related to the structure of the bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.

Keywords:  Real dynamics, topological entropy, hyperbolicity.
Mathematics Subject Classification:  Primary: 37B40; Secondary: 37F10, 37F20.

Received: May 2006;      Revised: January 2007;      Published: June 2007.